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Homework Help: Deriving a thermodynamics relationship

  1. Sep 6, 2008 #1
    1. The problem statement, all variables and given/known data
    Derive:
    [tex] \left(\frac{\partial C_{P}}{\partial P}\right)_{T} = -T \left(\frac{\partial^2 V}{\partial T^2}\right)_{P} [/tex]


    2. Relevant equations
    I have already derived that
    [tex] C_{P} = C_{V} + \left[\left(\frac{\partial E}{\partial V}\right)_{T} + P \right] \left( {\frac{\partial V}{\partial T}\right)_{P} [/tex]

    I also know that
    [tex]C_{V} = \left( \frac{\partial E}{\partial T}\right)_{V} [/tex]

    I also know that you can write differentials like this:
    [tex] z = z(x,y) [/tex]
    [tex] dz = \left( \frac{\partial z}{\partial x}\right)_{y} dx + \left( \frac{\partial z}{\partial y}\right)_{x} dy [/tex]

    3. The attempt at a solution
    My problem is not knowing what variables [tex] C_{P} [/tex] is a function of. Therefore, I do not know how to write its differential (if you even can). I also do not know how to differentiate the above expression for [tex] C_{P} [/tex] with respect to P holding T constant.

    I also do not know how to use Euler's chain rule for something like [tex] C_{P} [/tex] because i've never messed with anything that had more than 2 independent variables.

    I have tried three pages worth of manipulating, but I do not know if i'm even manipulating correctly because of what i mentioned above.

    Anything to get me started or tell me what is legal.. would be greatly appreciated..
     
  2. jcsd
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